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Chapter 10: Comparing Two Populations or GroupsSection 10.1Comparing Two Proportions

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IntroductionSuppose we want to compare the proportions of individuals with a certain characteristic in Population 1 and Population 2. Let’s call these parameters of interest p 1 and p 2. The ideal strategy is to take a separate random sample from each population and to compare the sample proportions with that characteristic.

What if we want to compare the effectiveness of Treatment 1 and Treatment 2 in a completely randomized experiment? This time, the parameters p 1 and p 2 that we want to compare are the true proportions of successful outcomes for each treatment. We use the proportions of successes in the two treatment groups to make the comparison. Here’s a table that summarizes these two situations.

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The Sampling Distribution of a Difference Between Two Proportions

In Chapter 7, we saw that the sampling distribution of a sample proportion has the following properties:

Shape: Approximately Normal if np ≥ 10 and n(1 – p) ≥ 10

ˆCenter: p p

ˆ

(1 )Spread: if the sample is no more than 10% of the population p

p p

n

To explore the sampling distribution of the difference between two proportions, let’s start with two populations having a known proportion of successes.

At School 1, 70% of students did their homework last night At School 2, 50% of students did their homework last night.

Suppose the counselor at School 1 takes an SRS of 100 students and records the sample proportion that did their homework.

School 2’s counselor takes an SRS of 200 students and records the sample proportion that did their homework.

1 2ˆ ˆWhat can we say about the difference in the sample proportions?p p

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The Sampling Distribution of a Difference Between Two Proportions

Using Fathom software, we generated an SRS of 100 students from School 1 and a separate SRS of 200 students from School 2. The difference in sample proportions was then calculated and plotted. We repeated this process 1000 times. The results are below:

1 2

What do you notice about the shape, center, and spread

ˆ ˆof the sampling distribution of ?p p

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The Sampling Distribution of a Difference Between Two Proportions

1 2 1 2ˆ ˆ ˆ ˆBoth and are random variables. The statistic is the difference

of these two random variables. In Chapter 6, we learned that for any two

independent random variables and ,

X Y X Y

p p p p

X Y

2 2 2 and X Y X Y

1 2 1 2ˆ ˆ ˆ ˆ 1 2

Therefore,

p p p p p p

ˆ p 1 ˆ p 2

2 ˆ p 1

2 ˆ p 2

2

p1(1 p1)

n1

2

p2(1 p2)

n2

2

p1(1 p1)

n1

p2(1 p2)

n2

ˆ p 1 ˆ p 2 p1(1 p1)

n1

p2(1 p2)

n2

Choose an SRS of size n1 from Population 1 with proportion of successesp1 and an independent SRS of size n2 from Population 2 with proportion ofsuccesses p2.

The Sampling Distribution of the Difference Between Sample Proportions

1 2 The mean of the sampling distribution is . That is,

the difference in sample proportions is an unbiased estimator of

the difference in population propotions.

p pCenter

1 1 1 1 2 2 2 2

1 2

When , (1 ), and (1 ) are all at least 10, the

ˆ ˆsampling distribution of is approximately Normal.

n p n p n p n p

p p

Shape

1 2

1 1 2 2

1 2

ˆ ˆ The standard deviation of the sampling distribution of is

(1 ) (1 )

as long as each sample is no more than 10% of its population

p p

p p p p

n n

Spread

(10% condition).

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Example 1: Suppose that there are two large high schools, each with more than 2000 students, in a certain town. At School 1, 70% of students did their homework last night. Only 50% of the students at School 2 did their homework last night. The counselor at School 1 takes an SRS of 100 students and records the proportion that did homework. School 2’s counselor takes an SRS of 200 students and records the proportion that did homework. School 1’s counselor and School 2’s counselor meet to discuss the results of their homework surveys. After the meeting, they both report to their principals that

1 2ˆ ˆ = 0.10.p p

1 2ˆ ˆa) Describe the shape, center, and spread of the sampling distribution of .p p

1 1 1 1 2 2

2 2

1 2

Because = 100(0.7) =70, (1 ) 100(0.30) 30, = 200(0.5) =100

and (1 ) 200(0.5) 100 are all at least 10, the sampling distribution

ˆ ˆof is approximately Normal.

n p n p n p

n p

p p

1 2Its mean is 0.70 0.50 0.20.p p

Its standard deviation is

0.7(0.3) 0.5(0.5)0.058.

100 200

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b) Find the probability of getting a difference in sample proportions

ˆ ˆ of 0.10 or less from the two surveys.p p

1 2ˆ ˆStandardize: When 0.10,

0.10 0.201.72

0.058

p p

z

c) Does the result in part (b) give us reason to doubt

the counselors' reported value?

There is only about a 4% chance of getting a difference in sample proportions

as small as or smaller than the value of 0.10 reported by the counselors.

This does seem suspicious!

Use Table A: The area to the left of 1.72

under the standard Normal curve is 0.0427.

z

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Confidence Intervals for p 1 – p 2

1 2 1 2

When data come from two random samples or two groups in a randomized

ˆ ˆexperiment, the statistic is our best guess for the value of . We

can use our familiar formula to calculate a confi

p p p p

1 2dence interval for :p p

statistic ± (critical value) × (standard deviation of statistic)

1 2

1 2

1 1 2 2ˆ ˆ

1 2

When the Independent condition is met, the standard deviation of the statistic

ˆ ˆ is:

(1 ) (1 ) p p

p p

p p p p

n n

1 2

1

Because we don’t know the values of the parameters and , we replace them

in the standard deviation formula with the sample proportions. The result is the

ˆ ˆstandard error of the statistic

p p

p p 1 1 2 22

1 2

ˆ ˆ ˆ ˆ(1 ) (1 ):

p p p p

n n

If the Normal condition is met, we find the critical value z* for the given confidencelevel from the standard Normal curve. Our confidence interval for p1 – p2 is:

1 1 2 21 2

1 2

statistic (critical value) (standard deviation of statistic)

ˆ ˆ ˆ ˆ(1 ) (1 )ˆ ˆ( ) *

p p p pp p z

n n

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Two-Sample z Interval for p 1 – p 2

Two-Sample z Interval for a Difference Between Proportions

1 2

1 1 2 21 2

1 2

When the Random, Normal, and Independent conditions are met, an approximate level C

ˆ ˆconfidence interval for ( ) is

ˆ ˆ ˆ ˆ(1 ) (1 )ˆ ˆ ( ) *

where

p p

p p p pp p z

n n

z

* is the critical value for the standard Normal curve with area C between * and *.z z

Random: The data are produced by a random sample of size n1 from Population 1 and a random sample of size n2 from Population 2 or by two groups of size n1 and n2 in a randomized experiment.

1 1 1 1 2 2 2 2

The counts of "successes" and "failures" in each sample or group --

ˆ ˆ ˆ ˆ, (1 ), and (1 ) -- are all at least 10.n p n p n p n p Normal:

Both the samples or groups themselves and the individual observations

in each sample or group are independent. When sampling without replacement, check

that the two populations are at least 10 times as large as the corresponding samples

(the 10% condition).

Independent:

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Plan: We should use a two-sample z interval for p1 – p2 if the conditions are satisfied.

Random The data come from a random sample of 800 U.S. teens and a separate random sample of 2253 U.S. adults.

Normal We check the counts of “successes” and “failures” and note the Normal condition is met since they are all at least 10:

Independent We clearly have two independent samples—one of teens and one of adults. Individual responses in the two samples also have to be independent. There are at least 10(800) = 8000 U.S. teens and at least 10(2253) = 22,530 U.S. adults.

Example 2: As part of the Pew Internet and American Life Project, researchers conducted two surveys in late 2009. The first survey asked a random sample of 800 U.S. teens about their use of social media and the Internet. A second survey posed similar questions to a random sample of 2253 U.S. adults. In these two studies, 73% of teens and 47% of adults said that they use social-networking sites. Use these results to construct and interpret a 95% confidence interval for the difference between the proportion of all U.S. teens and adults who use social-networking sites.

n1ˆ p 1 = 800(0.73) = 584 n1(1 ˆ p 1) 800(1 0.73) 216

n2ˆ p 2 = 2253(0.47) =1058.91 1059 n2(1 ˆ p 2) 2253(1 0.47) 1194.09 1194

State: Our parameters of interest are p1 = the proportion of all U.S. teens who use social networking sites and p2 = the proportion of all U.S. adults who use social-networking sites. We want to estimate the difference p1 – p2 at a 95% confidence level.

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Do: Since the conditions are satisfied, we can construct a two-sample z interval for the difference p1 – p2.

Conclude: We are 95% confident that the interval from 0.223 to 0.297 captures the true difference in the proportion of all U.S. teens and adults who use social-networking sites.

1 1 2 21 2

1 2

ˆ ˆ ˆ ˆ(1 ) (1 ) 0.73(0.27) 0.47(0.53)ˆ ˆ( ) * (0.73 0.47) 1.96

800 2253

0.26 0.037

(0.223, 0.297)

p p p pp p z

n n

AP EXAM TIP You may use your calculator to compute a confidence interval on the AP Exam. But there’s a risk involved. If you give just the calculator answer with no work, you’ll get either full credit for the “Do” step (if the interval is correct) or no credit (if it’s wrong). If you opt for the calculator method, be sure to name the procedure (e.g., two-proportion z interval) and to give the interval (e.g., 0.223 to 0.297).

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Significance Tests for p 1 – p 2

If the Random, Normal, and Independent conditions are met, we can proceed with calculations.

An observed difference between two sample proportions can reflect an actual difference in the parameters, or it may just be due to chance variation in random sampling or random assignment. Significance tests help us decide which explanation makes more sense. The null hypothesis has the general form

H0: p1 – p2 = hypothesized value

We’ll restrict ourselves to situations in which the hypothesized difference is 0. Then the null hypothesis says that there is no difference between the twoparameters:

H0: p1 – p2 = 0 or, alternatively, H0: p1 = p2

The alternative hypothesis says what kind of difference we expect.

Ha: p1 – p2 > 0, Ha: p1 – p2 < 0, or Ha: p1 – p2 ≠ 0

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Example 3: Researchers designed a survey to compare the proportions of children who come to school without eating breakfast in two low-income elementary schools. An SRS of 80 students from School 1 found that 19 had not eaten breakfast. At School 2, an SRS of 150 students included 26 who had not had breakfast. More than 1500 students attend each school. Do these data give convincing evidence of a difference in the population proportions? State appropriate hypotheses for a significance test to answer this question. Define any parameters you use.

Our hypotheses are

H0: p1 – p2 = 0 OR H0: p1 = p2

Ha: p1 – p2 ≠ 0Ha: p1 ≠ p2

where p1 = the true proportion of students at School 1 who did not eat breakfast, and p2 = the true proportion of students at School 2 who did not eat breakfast.

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1 2

1 2

ˆ ˆTo do a test, standardize to get a statistic:

statistic parameter test statistic

standard deviation of statistic

ˆ ˆ( ) 0

standard

p p z

p pz

deviation of statistic

If H0: p1 = p2 is true, the two parameters are the same. We call their common value p. But now we need a way to estimate p, so it makes sense to combine the data from the two samples. This pooled (or combined) sample proportion is:

1 2

1 2

count of successes in both samples combinedˆ

count of individuals in both samples combinedC

X Xp

n n

1 2

1 2

1 2

ˆUse in place of both and in the expression for the denominator of the test

statistic:

ˆ ˆ( ) 0

ˆ ˆ ˆ ˆ(1 ) (1 )

C

C C C C

p p p

p pz

p p p p

n n

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If the following conditions are met, we can proceed with a two-sample z test for the difference between two proportions:

1

2

1 2

The data are produced by a random sample of size from

Population 1 and a random sample of size from Population 2 or by

two groups of size and in a randomized experiment.

n

n

n n

Random

1 1 1 1 2 2 2 2

The counts of "successes" and "failures" in each sample or

ˆ ˆ ˆ ˆgroup -- , (1 ), and (1 ) -- are all at least 10.n p n p n p n p Normal

Both the samples or groups themselves and the individual

observations in each sample or group are independent. When sampling

without replacement, check that the two populations are at least

Independent

10 times

as large as the corresponding samples (the 10% condition).

Two-Sample z Test for the Difference Between Proportions

0 1 2

Suppose the Random, Normal, and Independent conditions are met. To

ˆtest the hypothesis : 0, first find the pooled proportion of

successes in both samples combined. Then compute the statiCH p p p

z

1 2

1 2

stic

ˆ ˆ( ) 0

ˆ ˆ ˆ ˆ(1 ) (1 )

Find the -value by calculating the probabilty of getting a statistic this

large or larger in the direction specified

C C C C

p pz

p p p p

n n

P z

by the alternative hypothesis :aH

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Plan: We should perform a two-sample z test for p1 – p2 if the conditions are satisfied.

Random: The data were produced using two simple random samples–of 80 students from School 1 and 150 students from School 2.

Normal:

Independent We clearly have two independent samples—one from each school. Individual responses in the two samples also have to be independent. There are at least 10(80) = 800 students at School 1 and at least 10(150) = 1500 students at School 2.

Example 4: Researchers designed a survey to compare the proportions of children who come to school without eating breakfast in two low-income elementary schools. An SRS of 80 students from School 1 found that 19 had not eaten breakfast. At School 2, an SRS of 150 students included 26 who had not had breakfast. More than 1500 students attend each school. Do these data give convincing evidence of a difference in the population proportions? Carry out a significance test at the α = 0.05 level to support your answer.

1 1 1 1 2 2 2 2ˆ ˆ ˆ ˆ=19 10, (1 ) 61 10, =26 10, (1 ) 124 10n p n p n p n p

State: Our hypotheses are

H0: p1 – p2 = 0 OR H0: p1 = p2

Ha: p1 – p2 ≠ 0Ha: p1 ≠ p2

where p1 = the true proportion of students at School 1 who did not eat breakfast, and p2 = the true proportion of students at School 2 who did not eat breakfast.

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1 2

1 2

Test statistic:

ˆ ˆ( ) 0 (0.2375 0.1733) 01.17

ˆ ˆ ˆ ˆ(1 ) (1 ) 0.1957(1 0.1957) 0.1957(1 0.1957)80 150

C C C C

p pz

p p p p

n n

Do: Since the conditions are satisfied, we can perform a two-sample z test for the difference p1 – p2.

1 2

1 2

19 26 45ˆ 0.1957

80 150 230C

X Xp

n n

P-value Using Table A or normalcdf, the desired P-value is2P(z ≥ 1.17) = 2(1 - 0.8790) = 0.2420.

Conclude: We fail to reject H0. Since our P-value, 0.2420, is greater than α = 0.05. There is not enough evidence to suggest that the proportions of students at the two schools who didn’t eat breakfast are different.

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Example 5: High levels of cholesterol in the blood are associated with higher risk of heart attacks. Will using a drug to lower blood cholesterol reduce heart attacks? The Helsinki Heart Study recruited middle-aged men with high cholesterol but no history of other serious medical problems to investigate this question. The volunteer subjects were assigned at random to one of two treatments: 2051 men took the drug gemfibrozil to reduce their cholesterol levels, and a control group of 2030 men took a placebo. During the next five years, 56 men in the gemfibrozil group and 84 men in the placebo group had heart attacks. Is the apparent benefit of gemfibrozil statistically significant? Perform an appropriate test to find out.

State: Our hypotheses areH0: p1 – p2 = 0 OR H0: p1 = p2

Ha: p1 – p2 < 0 Ha: p1 < p2

where p1 is the actual heart attack rate for middle-aged men like the ones in this study who take gemfibrozil, and p2 is the actual heart attack rate for middle-aged men like the ones in this study who take only a placebo. No significance level was specified, so we’ll use α = 0.01.

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1 2

1 2

Test statistic:

ˆ ˆ( ) 0 (0.0273 0.0414) 02.47

ˆ ˆ ˆ ˆ(1 ) (1 ) 0.0343(1 0.0343) 0.0343(1 0.0343)2051 2030

C C C C

p pz

p p p p

n n

Do: Since the conditions are satisfied, we can perform a two-sample z test for the difference p1 – p2.

1 2

1 2

56 84ˆ

2051 2030

1400.0343

4081

C

X Xp

n n

P-value Using Table A or normalcdf, the desired P-value is 0.0068

Conclude: Reject H0. Since the P-value, 0.0068, is less than α = 0.01. There is enough evidence to suggest that there is a lower heart attack rate for middle-aged men like these who take gemfibrozil than for those who take only a placebo.

Plan: We should perform a two-sample z test for p1 – p2 if the conditions are satisfied.

Random The data come from two groups in a randomized experiment

Normal The number of successes (heart attacks!) and failures in the two groups are 56, 1995, 84, and 1946. These are all at least 10, so the Normal condition is met.

Independent Due to the random assignment, these two groups of men can be viewed as independent. Individual observations in each group should also be independent: knowing whether one subject has a heart attack gives no information about whether another subject does.